Department of Mathematics
Server

An abstract configuration is said to be geometrically realizable if the points and lines are the usual points and lines in the real plane. It has been recently proved that there are no geometric configurations \( (n_4) \) for \( n \le 17 \). Here we present a new algebraic procedure to embed the cyclic \( (n_4) \) configurations in groups such that whenever four points \( \{P,Q,R,S\} \) form a block in the \( (n_4) \)-configuration, then \( P+Q+R+S = 0 \) in the group. Using this group law, we obtain a new geometric realization of the \( (n_4) \)s over the real plane where the four points in a block are now concyclic. In particular, we prove that cyclic \( (n_4) \)-configurations \( C_4(n,1,4,6) \) are circle-realizable in the real plane for several values of \( n \) including the "missing" cases \( n = 15, 16 \mbox{ and }17\). This is a part of an ongoing research done in collaboration with my student Jeff Lanyon.

References:

1. Branko Grünbaum, Which \( (n_4)\)s exist?, Geombinatorics 9 (2000), no. 4, 164–169.
2. Bokowski, Jürgen; Grünbaum, Branko; Schewe, Lars, Topological configurations exist for all \( n\ge17 \). European J. Combin. 30 (2009), no. 8, 1778–1785.
3. [3] Bokowski, Jürgen; Schewe, Lars, On the finite set of missing geometric configurations \( (n_4) \). Comput. Geom. 46 (2013), no. 5, 532–540.